how to change a mesh given by mathematical function with time i.e. how to include the time parameter in the mathematical equation?
I would suggest using Animation Nodes for more interactive results.

The Grid Mesh node generates evenly spaced vectors (Which can act as your 2D coordinates plane).



I separate the vectors into their x,y,z components (Z is zero in the grid), for every vector I compute the equation you provided above in terms of the x and y values I extracted, then I form a new vector from the x, y and z where z is the value we computed, resulting in points that satisfy your function:



In order to make it a 3D mesh with faces, we use the original polygon indices and new vectors to make a mesh as follows, this is just how meshed are output to object:



Last step is to introduce the (t) in your equation, which will be time, we can use the animate float node with the starting and ending values -1, 1 respectively as you stated in the question. And of course Duration is the duration of the animation. Resulting in the animation as you see:



If you want to color the surface based on its z value, where the higher the point is the higher the red value and the slower it is the bluer it is, you can use the following shading node tree: